Letter to Carl H. Brans: Question on Einstein’s Field Equations

Dear Dr. Carl Brans,

First, thank you for your outstanding contributions to modern physics.

I have a technical question about Einstein’s field equations for the static spherically symmetric nonvacuum, and I was hoping you might be able to provide the answer. As an independent researcher, I am working on a modified gravity theory which critically depends on the results.

My calculations show that, in the above case, Einstein’s field equations reduce to:

κρ(r) = -1/r2 – 1/r2g11 + g11′/g112r

κp(r) = -1/r2 – 1/r2g11 – g00′/g00g11r

where ρ is density, p is pressure, κ=-8πG/c2 and primes denote derivatives with respect to r. However, an equivalent pair of equations (in different notation using opposite metric signature and opposite sign for κ) are also presented in a paper by Zaregonbadi, Farhoudi, and Riazi (Phys. Rev. D 94, 084052 (2016)), in which they have a minus sign in front of κp(r) on the left-hand side of the second equation (in my notation):

κρ(r) = -1/r2 – 1/r2g11 + g11′/g112r

-κp(r) = -1/r2 – 1/r2g11 – g00′/g00g11r

I need to know which sign is correct. (My calculations were extensive, and finding a sign error would be nearly impossible.)

One clue to the answer is as follows: Using the equations of Zaregonbadi et. al., one would need an Equation of State ρ = -p to obtain the Schwarzschild-de Sitter solution, in which g00 is the negative inverse of g11.. However, using my equations, one would need an EoS ρ = p, which I understand is more physical, to get the same SdS solution. If you could verify the right EoS for me, that would also answer my question.

I consider you the most qualified person I can think of to clarify this detail. I greatly appreciate your time.